Supplemental Material Geometric Sequences

Motivating Examples Suppose a business makes a $1,000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its 12th month? How much profit will it earn in its first year? An interest-free loan of $12,000 requires monthly payments of 15% of the unpaid balance. What is the unpaid or outstanding balance after 18 payments? Problems like these can be solved using geometric sequences.

Introductory Example Find how much $5,000 grows to when it is invested at 8% annual compound interest. Beginning Balance = $5,000 Balance at end of year 1 = $5,000 x 1.08 = $5,400 Balance at end of year 2 = $5,400 x 1.08 = $5,832 Balance at end of year 3 = $5,832 x 1.08 = $6,298.56 Balance at end of year 4 = $6,298.56 x 1.08 = $6,802.44 Balance at end of year 5 = $6,802.44 x 1.08 = $7,346.64

Introductory Example Continued The sequence of numbers $5,000, $5,400, $5,832, $6,298.56, $6,802.44, $7,346.64 is called a geometric sequence. Each number, or term, is 1.08 times the previous term. The compounding of interest is essentially a geometric sequence. A geometric sequence is a sequence of numbers, called terms, such that each term is a constant multiple of the proceeding term. Any two consecutive numbers in the sequence are separated by a constant multiple called a fixed common ratio. The common ratio is equal to r. (r= 1.08 in the above sequence since $5,400/$5,000 = 1.08)

Geometric Sequence Examples Find the common ratio in the sequence 1, 3, 9, . . . r = 3/1 = 3 Find the next 3 terms of the sequence 1, 3, 9, . . . 27, 81, 243 Find the common ratio in the sequence 4, 2, 1, . . . r = 2/4 = ½ Find the next 3 terms of the sequence 4, 2, 1, . . . 1/2, 1/4, 1/8

Generalized Way to Write a Geometric Sequence A geometric sequence can be written as a0, a1, a2, a3, a4, + . . . a0, a0r, a0r2, a0r3, a0r4, . . . a0 = first term in the sequence a1 = a0r = second term in the sequence a2 = a0r2 = third term in the sequence a3 = a0r3 = fourth term in the sequence The n+1st term of a geometric sequence is an = a0 rn a0 is the first term in the sequence r is the common ratio (r = a1/a0)

Examples Finding the nth Term Find the nth term of the geometric sequence 1, 4, 16, 64, . . . an = a0 rn a0 = 1 and r = 4/1 = 4 an = 1 x 4 n = 4 n Find the 6th term of the geometric sequence 1, 4, 16, 64, . . . The 6th term of the sequence is a5. an = 4 n a5 = 4 5 = 1,024

Geometric Sequences and Compound Interest The compounding of interest is essentially a geometric sequence. geometric sequence compound interest formula an = a0 rn FV = PV(1 + i)n common ratio = r = 1 + i initial value = a0 = PV Consider again the sequence of numbers $5,000, $5,400, $5,832, $6,298.56, $6,802.44, $7,346.64 that comes from investing $5,000 at 8% annually compounded interest. an = $5,000 (1.08)n and FV = $5,000 (1 + .08)n

Discrete and Continuous Functions A geometric sequence is a discrete set of points. This discrete set of points forms a continuous exponential function if you connect the points. This exponential function can be written as y = a bx where a = initial value and b = common ratio. Continuous compounding exhibits exponential growth. an y = 5,000(1.08)x n

Application Example 1 Suppose a business makes a $1,000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its 12th month? Solution Month 1 : $1,000 Month 2 : $1,000 + (0.1)( $1,000) = $1,000(1.1) Month 3: $1,000(1.1)2 Find the 12th term (a11) in the geometric sequence a0 = $1,000 r = 1.1 (this comes from 100% + 10% or 1 + i) an = a0 rn a11 = $1,000 (1.1)(11) = $2,853.12

Graph for Application Example 1 an = a0 rn a0 = $1,000 and r = 1.1 an = $1,000 (1.1)n an y = $1,000(1.1)x n

Application Example 2 If changing market conditions cause a company earning $8,000,000 in 2010 to project a loss of 2% of its profit in each of the next 4 years, what profit does it project for 2014? Solution 2010: $8,000,000 2011 : $8,000,000 -0.02($8,000,000) = $8,000,000(0.98) 2012: $8,000,000(0.98)2 Find the 5th term (a4) in the geometric sequence a0 = $8,000,000 r = 0.98 an = a0 rn a4 = $8,000,000 (0.98)4 = $7,378,945.28

Application Example 3 An interest-free loan of $12,000 requires monthly payments of 10% of the outstanding balance. What is the outstanding balance after 18 payments? Solution When 10% of the outstanding balance is paid each month, 90% of the balance is left. We will assume the payment is made at the end of the month. Balance at the beginning of the first month : $12,000 Balance at the beginning of the second month: $12,000(0.9) Balance at the beginning of the third month: $12,000(0.9)2 We need to find the balance at the beginning of the 19th month an = a0 rn a18 = $12,000 (0.9)18 = $1,801.14

Graph for Application Example 3 Unpaid Balance This graph shows the unpaid balance and when the loan will be paid off. y = $12,000(0.9)x Months

Sum of the First n Terms of a Geometric Sequence The first n terms of a geometric sequence can be written from a0 to an-1 as a0, a0r, a0r2, a0r3 , . . . , a0 rn-1 We let Sn represent the sum of the first n terms on the sequence (1) Sn = a0 + a0r + a0r2 + a0r3 + . . . + a0 rn-1 If we multiply equation (1) by r, we have (2) rSn = a0r + a0r2 + a0r3 + a0r4 + . . . + a0 rn Subtracting equation (2) form equation (1), we obtain Sn – rSn = a0 + (a0r – a0r) + (a0r2 – a0r2 ) + (a0 rn-1- a0 rn-1) – a0 rn Thus Sn (1 – r) = a0 – a0 rn Sn = a0 (1 - rn)/(1 – r)

Sum of the First n Terms of a Geometric Sequence The sum of the first n terms of a geometric sequence with first term a0 and common ratio r is This formula only works if r is not equal to 1.

Application Example 4 Suppose a business makes a $1,000 profit in its first month and has its profit increase by 10% each month for the next 2 years. How much profit will the business earn in its first year? Solution The profit it will earn in its first year is the sum of the profit in the first twelve months. Sn = a0 (1 – rn)/(1 – r) n = 12 a0 = $1,000 r = 1 + i = 1.1 S12 = $1,000(1 - (1.1)12)/(1 – 1.1) S12 = $21,384.28 NOTE: You do include the first term when finding the sum so, you are summing up 12 numbers!